Illustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited

The inverse square force law admits a conserved vector that lies in the plane of motion. This vector has been associated with the names of Laplace, Runge, and Lenz, among others. Many workers have explored aspects of the symmetry and degeneracy associated with this vector and with analogous dynamical symmetris. We define a conserved dynamical variable $\alpha$ that characterizes the orientation of the orbit in two-dimensional configuration space for the Kepler problem and an analogous variable $\beta$ for the isotropic harmonics oscillator. This orbit orientation variable is canonically conjugate to the angular momentum component normal to the plane of motion. We explore the canoncial one-parameter group of transformations generated by $\alpha (\beta).$ Because we have an obvious pair of conserved canonically conjugate variables, it is desirable to us them as a coordinate-momentum pair. In terms of these phase space coordinates, the form of the Hamiltonian is nearly trivial because neither member of the pair can occur explicitly in the Hamiltonian. From these considerations we gain a simple picture of the dynamics in phase space. The procedure we use is in the spirit of the Hamilton-Jacobi method.Comment: 15 pages, 1 figure, to be published in American Journal of Physic

Flux Attractors and Generating Functions

We use the flux attractor equations to study IIB supergravity compactifications with ISD fluxes. We show that the attractor equations determine not just the values of moduli fields, but also the masses of those moduli and the gravitino. We then show that the flux attractor equations can be recast in terms of derivatives of a single generating function. We also find a simple expression for this generating function in terms of the gravitino mass, with both quantities considered as functions of the fluxes. For a simple prepotential, we explicitly solve the attractor equations. We conclude by discussing a thermodynamic interpretation of this generating function, and possible implications for the landscape.Comment: 43 pages. v2, references added and typos correcte

RascalC: A Jackknife Approach to Estimating Single and Multi-Tracer Galaxy Covariance Matrices

To make use of clustering statistics from large cosmological surveys, accurate and precise covariance matrices are needed. We present a new code to estimate large scale galaxy two-point correlation function (2PCF) covariances in arbitrary survey geometries that, due to new sampling techniques, runs $\sim 10^4$ times faster than previous codes, computing finely-binned covariance matrices with negligible noise in less than 100 CPU-hours. As in previous works, non-Gaussianity is approximated via a small rescaling of shot-noise in the theoretical model, calibrated by comparing jackknife survey covariances to an associated jackknife model. The flexible code, RascalC, has been publicly released, and automatically takes care of all necessary pre- and post-processing, requiring only a single input dataset (without a prior 2PCF model). Deviations between large scale model covariances from a mock survey and those from a large suite of mocks are found to be be indistinguishable from noise. In addition, the choice of input mock are shown to be irrelevant for desired noise levels below $\sim 10^5$ mocks. Coupled with its generalization to multi-tracer data-sets, this shows the algorithm to be an excellent tool for analysis, reducing the need for large numbers of mock simulations to be computed.Comment: 29 pages, 8 figures. Accepted by MNRAS. Code is available at http://github.com/oliverphilcox/RascalC with documentation at http://rascalc.readthedocs.io

Flux Attractors and Generating Functions.

We use the flux attractor equations to study IIB supergravity compactifications with 3-form fluxes. We show that the attractor equations determine not just the values of the complex structure moduli and the axio-dilaton, but also the masses of those moduli and the gravitino. We then show that the flux attractor equations can be recast in terms of derivatives of a single generating function. A simple expression is given for this generating function in terms of the D3 tadpole and gravitino mass, with both quantities considered as functions of the fluxes. For a simple prepotential, we explicitly solve the attractor equations. We also discuss a thermodynamic interpretation of this generating function, and possible implications for the landscape. Having solved the flux attractor equations for 3-form fluxes, we add generalized fluxes to the compactifications and study their effects. We find that when we add only geometric fluxes, the compactifications retain their no-scale structure, and minimize their scalar potential when the appropriate complex flux is imaginary self-dual (ISD). These minima are still described by a set of flux attractor equations, which can be integrated by a generating function. The expressions for the vector moduli are formally identical to the case with 3-form fluxes only, while some of the hypermoduli vii are determined by extremizing the generating function. We work out several orbifold examples where all vector moduli and many hypermoduli are stabilized, with VEVs given explicitly in terms of fluxes.Ph.D.PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/75824/1/rcoconne_1.pd